Horizon-Aware Scheduling (HAS)

• HAS is a scheduling framework that explicitly embeds a finite planning horizon to guide constraints, objectives, and algorithmic updates.
• It improves system performance by incorporating horizon-dependent factors and bidirectional information flows, leading to robust outcomes in uncertain conditions.
• HAS employs dynamic programming, MILP, and reinforcement learning techniques, achieving significant performance gains in domains such as traffic control and multi-agent coordination.

Horizon-Aware Schedule (HAS) denotes a suite of scheduling, decision, and control methodologies in which explicit knowledge or estimation of a finite planning horizon fundamentally shapes algorithmic structure and information flows. Originating independently across optimization, control, and machine learning, HAS frameworks extend classical rolling-horizon and myopic schemes by incorporating horizon-dependent constraints, objectives, and coordination mechanisms. This enables improved system-level performance, adaptivity, and robustness in dynamic or uncertain environments.

1. Fundamental Principles and Formulations

At its core, a Horizon-Aware Schedule is characterized by the explicit encoding of a planning or resource allocation horizon $H$ (finite, receding, or uncertain) into the structure of constraints, objectives, and algorithmic updates. HAS appears across multiple domains, with key instantiations including:

• Rolling- and finite-horizon task allocation, in which assignment variables, resource consumption, or delay penalties are constrained or weighted over a sliding window of length $H$, as in MILP or MDP formulations for scheduling and queue control (Emam et al., 2020, Ayan et al., 2020, Spratt et al., 2018, Hu et al., 2019).
• Horizon-aware information exchange, wherein agents communicate forward and backward anticipated effects of scheduling decisions, with flows (such as congestion signals) tied to predicted horizon-bound effects (Hu et al., 2019).
• Horizon-parameterized control or policy learning: e.g., policies $\pi(a|s,g,h)$ conditioned not just on state-goal but also on remaining horizon $h$, with reachability or accessibility values $C^*(s,a,g,h)$ estimated over variable horizons (Naderian et al., 2020).
• Horizon-robust online resource allocation, reflecting optimization under uncertainty in the horizon $T$, via explicit schedule sequences $(\lambda_1,\dots,\lambda_{\tau_2})$ designed for worst-case or data-driven trade-offs (Balseiro et al., 2022).
• Distributed and real-time scheduling in physical agents, including adaptive action chunking in VLA models, where sampling and dispatch schedules are made index- and horizon-aware for latency minimization (Lu et al., 19 Mar 2026).

All HAS methodologies share components of (i) horizon-indexed optimization (decision variables and costs/constraints explicit in $H$ or $h$), (ii) the capacity to shift or adjust the horizon adaptively (e.g., receding or rolling window), and (iii) mechanisms to encode horizon uncertainty.

2. Representative Algorithms and Information Flows

Distinct HAS instantiations manifest as structured algorithms tailored to respective domains:

a) Decentralized Traffic Scheduling (Bi-Directional HAS):

Each intersection $i$ solves a phase-scheduling problem over $H$0, optimizing

$H$1

subject to phasing, clearance, and horizon bounds. State and predicted outflows are communicated to downstream neighbors. Critically, congestion feedback (mean delay) is sent upstream, and local costs are augmented by downstream congestion

$H$2

which implements an implicit Lagrangian decomposition, biasing against overloading neighbors (Hu et al., 2019).

b) Receding Horizon Multi-Agent Scheduling:

Agents $H$3, tasks $H$4, and a planning horizon $H$5 frame a MILP: $H$6 with assignment and precedence constraints, solved in rolling increments $H$7, repeatedly updating with fresh state and outcome information (Emam et al., 2020).

c) Horizon-Aware Policy Learning:

In reinforcement learning, horizon-aware policies and accessibility functions are defined: $H$8 satisfying Bellman recurrences with monotonicity in $H$9. The schedule $\pi(a|s,g,h)$0 parameterizes a family of policies trading path speed vs. reliability (Naderian et al., 2020).

d) Resource Allocation under Horizon Uncertainty:

A HAS here is $\pi(a|s,g,h)$1 with $\pi(a|s,g,h)$2, providing per-period target consumptions. The sequence is optimized (by LP) to maximize worst-case expected reward ratio over all possible horizons $\pi(a|s,g,h)$3 (Balseiro et al., 2022).

3. Bidirectionality, Coordination, and Network-Level Optimality

Bi-directional information flows are a defining feature in advanced HAS variants, as in decentralized traffic signal control:

• Forward (demand) flow: Predicted outflows encapsulate future demand for downstream neighbors; these are appended to local input queues.
• Backward (congestion) flow: Congestion signals, quantifying the downstream agent's delay, are propagated upstream and directly incorporated in local delay costs.
• This structure implements a local proxy objective at each agent,

$\pi(a|s,g,h)$4

which biases local optimization toward network-level efficiency. The congestion price acts analogously to a Lagrange multiplier in distributed optimization, yielding near-global optimality without centralized coordination (Hu et al., 2019).

A similar concept underpins horizon-robust online allocation, where schedules are optimized not just for expected horizon but for worst-case uncertainty, yielding competitive ratios with explicit information-theoretic lower bounds (Balseiro et al., 2022).

4. Computational Structure and Tractability

HAS formulations typically increase computational demands relative to purely myopic approaches, as the planning horizon $\pi(a|s,g,h)$5 multiplies the effective size of the decision space. Prominent computational mechanisms include:

• Dynamic Programming Recursions: For finite-horizon MDPs, HAS reduces to layered DP, where the state space grows with $\pi(a|s,g,h)$6; pruning by admissible actions (no-op for no fresh samples) yields significant practical complexity reduction (Ayan et al., 2020).
• Mixed Integer Programming (MIP)/Branch-and-Bound: Rolling-horizon MILPs are deployed for resource or task scheduling, where horizon truncation and load-balancing heuristics control tractability (Emam et al., 2020, Spratt et al., 2018).
• Convex/Linear Programming: In horizon-uncertain settings, LPs efficiently compute the best schedule of per-slot targets.
• Neural Approximation and RL: In domains where state or policy space is intractable, universal value function approximation and DQN are used to synthesize horizon-sensitive policies from synthetic experience (Naderian et al., 2020, Taga et al., 20 Mar 2026).

Empirically, careful pruning and tightly-coupled heuristics enable real-time or near-real-time solution of HAS problems for horizons $\pi(a|s,g,h)$7–$\pi(a|s,g,h)$8 in many practical domains (Ayan et al., 2020, Spratt et al., 2018, Emam et al., 2020).

5. Domain-Specific Instantiations and Empirical Results

Table: Central features and empirical benefits of HAS across domains.

Domain	HAS Instantiation	Observed Benefit
Urban traffic control	Bi-directional phase scheduling	15–30% delay reduction
Multi-agent search/rescue	Receding-horizon MILP	50% mission time reduction
Networked control systems	Pruned finite-horizon DP	Diminishing AoI/MSE for $\pi(a|s,g,h)$9
Operating theatre management	Rolling-horizon MIP/metaheuristic	Up to +11 patients/week
Online allocation under horizon uncertainty	Schedule sequence LP	Optimal competitive ratio
VLA model real-time inference	Index-dependent sampling schedule	2-3x lower TTFA/latency
Horizon-aware RL	$h$0-conditioned C-function/CAE	Speed-reliability trade-off
Anytime-valid testing	DQN-learned horizon-aware betting	Tightest CS, improved power

In decentralized traffic signal control, HAS with bi-directional exchange yields 15–30% average delay reduction under high congestion (Hu et al., 2019). In search and rescue, rolling-horizon scheduling cuts completion time by up to half versus myopic baselines (Emam et al., 2020). Finite-horizon AoI scheduling shows sharply diminishing returns past $h$1 despite exponential worst-case complexity, demonstrating the practical value of horizon truncation and action pruning (Ayan et al., 2020). Real-time robot control with HAS-based VLA sampling compresses time-to-first-action by $h$2–$h$3 on commodity GPUs and preserves trajectory quality (Lu et al., 19 Mar 2026).

6. HAS under Uncertainty and Adaptive Horizons

A major thread in HAS research addresses explicit horizon uncertainty:

• Uncertain or unknown horizons: HAS allocates resource schedules $h$4 for all $h$5 up to a maximum horizon $h$6, optimizing for the worst-case actual horizon $h$7 (Balseiro et al., 2022).
• Incorporating predictions: Schedules can interpolate between a robust schedule (for worst-case) and a predictive schedule (for estimated $h$8), with closed-form trade-off depending on a mixing parameter $h$9.
• Guarantees: Competitive ratio bounds $C^*(s,a,g,h)$0 are both algorithmically achievable and minimax optimal, even with adversarial horizon assignment.

This horizon-robust scheduling paradigm is central in online resource allocation, advertising, and control domains subject to workload variability and demand stochasticity.

7. Theoretical Guarantees, Extensions, and Impact

Horizon-Aware Scheduling approaches come with strong theoretical and empirical guarantees:

• Monotonicity in horizon: In RL, accessibility $C^*(s,a,g,h)$1 is provably non-decreasing in $C^*(s,a,g,h)$2; larger horizons yield at least as high success as smaller ones (Naderian et al., 2020).
• Optimality and decomposition: Bi-directional HAS structures admit decomposition into local objectives with global optima closely approximated via congestion-based coupling (Hu et al., 2019).
• Error bounds and phase diagrams: In anytime-valid testing, HAS delineates actionable phase regions (aggressive, Kelly, defensive) and trains DQN to near-theoretical optimality (Taga et al., 20 Mar 2026).
• Heuristic and metaheuristic integrations: In large-scale systems (e.g., elective surgery management), horizon-aware rolling-MIP is coupled with hyper metaheuristics for feasible large-instance solutions (Spratt et al., 2018).

HAS is extensible to new problem classes, with emerging applications in high-frequency robotics, continuous control, and online algorithms under non-stationarity. The unifying mathematical principle is the tight, explicit embedding of the planning horizon into all levels of schedule construction, resource allocation, and policy evaluation, providing a systematic basis for responsive and robust large-scale system optimization.